Nonuniqueness Sequences for Weighted Algebras of Holomorphic Functions in the Unit Circle

Abstract

Suppose that \(\mathcal{P}\) is a system of continuous subharmonic functions in the unit disk \(\mathbb{D}\) and \(\mathbb{A}_\mathcal{P}\) is the class of holomorphic functions f in \(\mathbb{D}\) such that log|f(z)| ≤ B f p f (z) + C f , z ∈ \(\mathbb{D}\), where B f and C f are constants and p f ∈ \(\mathcal{P}\). We obtain sufficient conditions for a given number sequence Λ = { λn} ⊂ \(\mathbb{D}\) to be a subsequence of zeros of some nonzero holomorphic function from \(\mathbb{A}_\mathcal{P}\), i.e., Λ is a nonuniqueness sequence for \(\mathbb{A}_\mathcal{P}\).